An optimization model for the integrated operation of the multiple reservoirs in the Upper Yellow River

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1 254 GIS and Remote Sensing in Hydrology, Water Resources and Environment (Proceedings of ICGRHWE held at the Three Gorges Dam. China. September 2003). IAHS Publ An optimization model for the integrated operation of the multiple reservoirs in the Upper Yellow River JIAQI HU 1, YANGBO CHEN 2 & SICHUN GAO 3 1 Department of Engineering Management, Three Gorges University, Yichang , China 2 Department of Water Resources and Environment, Sun Yat-Sen University, 135Xingangxi Road, Guangzhou, , China eescvb@zsu.edu.cn 3 Department of Hydrology and Water Resources, Wuhan University, Wuhan China Abstract The Yellow River is the second largest river in China. There are seven reservoirs and hydropower stations in the mainstream of the Upper Yellow River. This paper presents an optimal mathematical model for the integrated operation ofthe seven reservoirs and hydropower stations. The objective of this model is to maximize the sum of electricity generated by all the hydropower stations with the constraint of firm power, and the Discrete Differential Dynamic Programming (DDDP) is employed to find the optimal solution of this model. The decision variables of DDDP are chosen as the resetvoir discharge, while the state variables are reservoir storages, and the backward recursive equation is derived to find the reservoir optimal operation policies stage by stage. This model is used to find the integrated optimal operation policies of the past 33 years with obseived hydrological data, the results show that there has been a 5.89% increase in the average annual electricity generated by all the seven hydropower stations. This implies that the model presented in this paper is good and can be used to derive the integrated operating rules of the reservoirs in the Yellow River. Key words DDDP; optimization; reservoir operation; Yellow River INTRODUCTION The Yellow River is the second largest river in China, with a total length of 5464 km and a basin area of km 2. Seven reservoirs and hydropower stations have been built in the mainstream of the Upper Yellow River, which include Longyangxia, Lijiaxia, Liujiaxia, Yanguoxia, Bapanxia, Daxia and Qingtongxia. While Longyangxia and Liujiaxia have hydropower stations and huge reservoirs with long-term regulating capacities, the other five projects are mainly runoff hydropower stations with small regulating capacities. The total storage capacity of Longyangxia Reservoir and Liujiaxia Reservoir is billion m 3, the total installed capacity of the seven hydropower stations is MW. This paper presents an optimal mathematical model for the integrated operation of the seven reservoirs and hydropower stations. The objective of this model is to maximize the sum of electricity generated by all the hydropower stations with a constraint of firm power. The model is a non-linear mathematical model, and the Discrete Differential Dynamic Programming (DDDP) is employed to find the optimal solution of this model. This model is used to find the integrated optimal operation policies of the past 33 years

2 An optimization model for the multiple reservoirs in the Upper Yellow River 255 with observed hydrological data, the results show that there has been a 5.89% increase in the average annual electricity generated by all the seven hydropower stations. This shows that the model presented in this paper is good and can be used to derive the integrated operating rules of the reservoirs in the Upper Yellow River. THE MATHEMATICAL MODEL The model presented in this paper is an optimal mathematical model for the integrated operation of the seven reservoirs and hydropower stations of the Upper Yellow River. The objective function The objective of this model is to maximize the sum of electricity generated by all the hydropower stations with a constraint of firm power, which can be written as: T M r=l ;=1 NF > NFO (2) where i is reservoir and hydropower station numbers, and i = 1,2,3,4,5,6,7 represent Longyangxia, Lijiaxia, Liujiaxia, Yanguoxia, Bapanxia, Daxia and Qingtongxia, respectively; Mis the total number of reservoir and hydropower station, and M = 7; t is stage, and t= 1,2,,77, 77 is the total stages; E U T is the average generated electricity of z'-th hydropower station during stage t; NFO is the given firm power, and NF is the firm power, which can be determined as: f M NF = MN 2X f = l,2,,77 (3) where 7Y (., is the average generated power of z'-th hydropower station at stage t; There are two objective functions, according to Chen et. al (1996), the two objective functions can be integrated to form a single objective by combining Equations (1) and (2), the single objective function can be written as: MAX\Z [E, - a, A{EF - E, ) A ] (4) where E, is the total generated electricity of all the seven hydropower stations at stage t, which can be determined as M (5) where EF is the firm electricity, and EF = NFO At, where At is the stage length in hours, A is a fine coefficient, a is model parameter that can take the value of 2 to 5, and cj, is a 0-1 variable, which takes value as the following equation:

3 256 Yangbo Chen et al. E, > EFO c t =i.. E, < EFO (6) ''"Il The constraints There are several constraints that can be written as follows: (1) Water balance constraint = V u + (5,, + - Q u )At, i = 1,2,,7; t = 1,2,, T (7) Where V- Ut is the storage of z'-th reservoir at the beginning of stage t; Q u is the average discharge of i-th reservoir at stage t; S/j is the average inflows of z'-th reservoir at stage t; and x is the time of the discharge of the (z-l)-th reservoir flowing to the z'-th reservoir. (2) Reservoir storage constraint V U <VM U, z' = l,2,,7;? = 1,2, J (8) where VM i<t is the permitted maximum storage of z'-th reservoir at stage t. (3) Multiple utilization constraint V U >VN U, i = 1,2,,7;? = 1,2,,T (9) Where FA/,,, is the minimum storage of z'-th reservoir at stage t due to the multiple utilization. (4) Navigation constraint Q U >QN i = 1,2,,7;z = l,2,,t (10) where QNj is the minimum discharge of z'-th reservoir to guarantee downstream navigation. (5) Turbine discharge constraint Q Ll <QM i, / = 1,2,,7;/= 1,2,,T (11) where QM, is the permitted maximum discharge of the turbine of z'-th hydropower station. (6) Power constraint N U <NT :, i = 1,2,,l;t = 1,2,,T (12) where NT t is the maximum power that can be generated by z'-th hydropower station, which is determined by water head applied to the turbine. The mathematical model According to the above objective function and constraints, the mathematical model can be written as follows:

4 An optimization model for the multiple reservoirs in the Upper Yellow River 257 MAXY\E,-a,A(EF-EX\ V^= V u+{s u+ Q,_ u _ T -Q Ll )At s.t. VN U < V u < VM U (13) \QN i <Q u <QM l 0 < N u < NT, 1 = l,2,,7;t = l,2,,t This model is a nonlinear optimization model with 7 M T constraints. MODEL SOLUTION The Discrete Differential Dynamic Programming (DDDP; Heidari, 1971) is employed to find the optimal solution to this model. The DDDP model is set up as follows. Stage variables The stage variables are chosen as t. State variables Vijii = 1,3; t = 1,2,, 7) are chosen as the state variables of DDDP. As reservoir 2,4,5,6,7 have very small regulating capacities, so in this study, the value of V i<t (i = 2,4,5,6,7; t = 1,2,, 7) are assumed to be constant. The decision variables The decision variables of DDDP are chosen as the reservoir discharge Q,j(i = 1,3; t = 1,2,, 7). As the value of V U, (i = 2,4,5,6,7; / = 1,2,, 7) are assumed to be constant, so Q;,i(i = 2,4,5,6,7; / = 1,2,, I) are not variables, their values can be decided according to the following equation: Qu=S u +Q, / = 2,4,5,6,7; f = 1,2,. (14) Recursive equation The backward recursive equation is derived to find the model optimal operation policies stage by stage. The recursive equation is as follows: F; (V u, V u ) = MAX[R, + F; +x (V LL+L, V 3L+L )},t = T,T-1,,1 (15)

5 258 Yangbo Chen et al. Boundary conditions The boundary conditions include the followings: ^ + 1(^ r + 1.*3. r +i) = 0 (16) V u =V lj+i =VM x (17) Vyx=V^=VM, (18) where VM\ and VMi are the dead storage of reservoir 1 and 3, respectively. The initial result The initial result is very important in the calculation of DDDP. In this study, the initial result is selected from the past operation results, so it is a feasible solution. RESULTS The model and method for solving this model presented in this paper was used to study the integrated optimal operation of the seven reservoirs and hydropower stations in the Upper Yellow River and 33 years of hydrologie data was chosen to do this study. The stage length is 10 days so there are 1188 stages in total. The reservoir storage of reservoir 1 (Longyangxia) and 3 (Liuyangxia) in the whole 33 years are shown in Figs 1 and 2. The results show that there is a 5.89% increase in the average annual electricity generated by all the seven hydropower stations compared with the results derived by using the general method (Northwest Institute of Surveying and Design, 1998). From the optimal results, it has been found that a constraint corridor exists for the reservoir storage that limits the up and down limitation of the reservoir water storage at different stages. The corridors of Longyangxia and Liujiaxia are derived and shown in Figs 3 and I Fig. 1 Longyangxia Reservoir storage of the 33 years from 1957 to Most of the time the reservoir level was above 2580 m, and only in four very dry years was the reservoir level lower than 2560 m. Longyangxia Reservoir has multi-year regulating capacity, so the reservoir level was only low to the dead level in several dry years.

6 An optimization model for the multiple reservoirs in the Upper Yellow River Q 1 1 Fig. 2 Liujiaxia Reservoir storage of the 33 years from 1957 to The reservoir storage is above 1720 m most of the time. This is due to the regulation of Longyangxia, so that the Liujiaxia Reservoir can maintain a higher water head for the power generation. Without the Longyangxia Reservoir multi-year regulation, this can not be achieved. The benefit of integrated operation is acquired.

7 260 Yangbo Chen et al. CONCLUSIONS This paper presents an optimal optimization model for the integrated operation of the seven reservoirs and hydropower stations in the Upper Yellow River, and the Discrete Differential Dynamic Programming (DDDP) is employed to find the optimal solution of this model. The integrated optimal operation policies of the past 33 years were derived by using this model, and the results show that there has a 5.89% increase in the average annual electricity generated by all the seven hydropower stations compared with the result derived by using the general method. Acknowledgments This study was supported by the Northwest Power Dispatching Center. The authors thank Professor Hui-Yuan Chen, Senior Engineer Jiaoxin Zhou, Mr Jinhuai Xue for their beneficial discussions. REFERENCES Chen, Y. & Chen, A. (1996) Resen-oir optimal control: theory, method and applications. Science and Technology Publishing House of Hubei Province, Hubei, China (in Chinese). Hcidari, M. (1971) Discrete Differential Dynamic Programming approach to water resources system optimization. Water Resour. Res. 7(2), Northwest Institute of Surveying and Design (1998) Technical report for the integrated dispatching of Longyangxia and Liujiaxia by using the general method. Northwest Institute of Surveying and design. Xi'an, China (in Chinese).